# Questions tagged [mgf]

The moment generating function (mgf) is a real function which allows to derive a random variable's moments and therefore can characterize its entire distribution. Use also for its logarithm, the cumulant generating function.

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### Gamma Distribution satisfying property

How can we prove that gamma random variable $X_{n}$ with parameters $(n,3)$ can satisfy the following relation for some $n$? $$P(X_{n} < n/2) > 0.999$$ I used the definition of density function ...
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### Expectation of the product of polynomial & exponential transformations of normal r.v

Let $X \sim \mathcal{N}(\mu, \sigma^2)$. Are there any (1) general formula and (2) references to the general formula for $$\mathbb{E} (X^n e^{tX}),\; n \in \mathbb{N}, t \in \mathbb{R}$$ in ...
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### The log of a Moment Generating Function

I am having trouble understanding the rationale of the following related to the MGF: Function Mx(t)=E[exp(tX)], the expectation exists for all t in a neighbourhood of zero, and X has mean mx, show ...
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### How would a moment generating function change if all the random variables are increased by a value

Suppose you have some moment generating function $M_x(t)$ Now all the random variables x are increased by a arbitrary value b. What is the new moment generating value? I tried solving this by moving ...
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### Weak convergence of moment generating function

I have the following sequence of rvs $$Z_1 = X_0*Y_0$$ $$Z_{n+1} = Z_n /2 + X_n*Y_n$$ Where $X_n$ and $Y_n$ are independent, with $X_n$ having Bernoulli distribution with p=1/2 and $Y_n$ having ...
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### Inverting a moment generating function

If I have a random variable $X$ with pdf $f$ I can compute its MGF as $$M_X(t) = \int_{-\infty}^\infty e^{tx} f(x)dx.$$ My understanding is that this is basically a Laplace transformation. ...
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### Sum of indepedent random variables and a constant

Let $X_1$ and $X_2$ be independent Normal random variables with mean $\mu_1$ and $\mu_2$, and variances $\sigma_1$ and $\sigma_2$. Let $Y = X_2-X_1 + c$, where $c$ is a constant. For notational ...
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### How to prove there is no moment generating function for t distribution

I'm struggling to show that the moment generating function for t distribution does not exist. So far I tried to show the moment generating function diverges from its integration but the computation is ...
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### Moment generating function between 2 variables

I understand that mean of Y is M'Y(0), whereas variance of Y is M''Y(0). I can derive expressions through differentiation to get M'Y(0) and M''Y(0). The 2, however, have the expression Mx(0) in them. ...
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### How does MGF of Pareto distribution of first kind exist for non-positive values of t? [closed]

I have reached upto the stage shown in the attached picture. The r.v. X is always positive and its power $\beta+1$ is also always positive. Therefore, how can it be said that MGF exists for t <= 0? ...
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### In general, how should we find the pmf given only the moment generating function without comparing its form to that of famous pmf?

Background It is known that moment generating function generates moments, but does it hold information about the probability of the random variable being realised at a particular value? Example ...
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### When to prefer the moment generating function to the characteristic function?

Let $(\Omega, \mathcal{F}, P)$ be a probability space, and let $X : \Omega \to \mathbb{R}^n$ be a random vector. Let $P_X = X_* P$ be the distribution of $X$, a Borel measure on $\mathbb{R}^n$. The ...
Let $X_i:i=1,2,...,n$ be independent ~ $NegBin(\mu,\alpha)$random variables such that $E(X_i) = \mu$, $Var(X_i) = \mu + \frac{\mu^2}{\alpha}$. (i) Find the mean and variance of $Y=∑(X_i)$. (ii) Find ...
Suppose $\{X_n\}_{n\in \mathbb{N}}$ is a sequence of independent and identically distributed random variables and $S_n:=X_1+...+X_n$. Assume that each $X_i$ has mean $0$ and that all $X_i$ have a ...